This phase II trial is studying how well giving carboplatin, paclitaxel, cetuximab, and erlotinib hydrochloride together works in treating patients with metastatic or recurrent squamous cell head and neck cancer. Drugs used in chemotherapy, such as carboplatin and paclitaxel, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. Monoclonal antibodies, such as cetuximab, can block tumor growth in different ways. Some block the ability of tumor cells to grow and spread. Others find tumor cells and help kill them or carry tumor-killing substances to them. Erlotinib hydrochloride may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Giving combination chemotherapy together with cetuximab and erlotinib hydrochloride may kill more tumor cells.